On Generalized Nth -Order Recurrent Finsler Spaces Associated with Cartan’s Curvature Tensor

Abstract

This paper investigates the geometric structure of a Finsler space  admitting a generalized n^th -order recurrence property relative to Cartan’s third curvature tensor field . We define a new class of Finsler spaces, denoted as - , where the h-covariant derivative of the n^th order of the curvature tensor  satisfies a specific recurrence condition involving multiple recurrence tensor fields. Throughout this study, we derive several fundamental geometric identities, including Bianchi and Veblen-type identities, within the framework of this generalized space. Furthermore, we establish necessary and sufficient conditions under which Cartan’s fourth curvature tensor  and the Ricci tensor exhibit recurrence properties. The results demonstrate that the curvature scalar and the deviation tensor associated with Cartan’s third curvature tensor are non-vanishing in - . These findings extend the existing theory of recurrent Finsler spaces to higher-order differential manifolds.